Method For Projecting Out Irreducible Representations From a Quantum State of n Particles with d Colors

ABSTRACT

We describe a method for using a classical computer to generate a particular sequence of elementary operations (SEO), an instruction set for a quantum computer. Such a SEO will induce a quantum computer to perform a unitary transformation U that we call an Irreps Gen U. This U simultaneously diagonalizes a set of operators H μ  called HYPs (Hermitian Young Projectors) for n particles with d colors or, equivalently, for n qu(d)its. H μ  projects out n particle irrep μ of U(d).

CROSS REFERENCES TO RELATED APPLICATIONS

This patent application makes reference to patent application Ser. No.14/499,539 by R. R. Tucci entitled: “Method For Projecting OutIrreducible Representations From a Quantum State Of Multiple Qubits”.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH AND DEVELOPMENT

Not Applicable

REFERENCE TO COMPUTER PROGRAM LISTING

A computer program listing consisting of a single file entitled:

-   -   IrrepsGen-d-colors-maxima.txt,        in ASCII format, is included with this patent application. The        file is a collection of Maxima commands which do most of the        labor-intensive calculations underpinning this invention. Maxima        is a free, open source Symbolic Manipulation Program (SMP).

BACKGROUND OF THE INVENTION

(A) Field of the Invention

The invention relates to a quantum computer; that is, an array ofquantum bits (called qubits). More specifically, it relates to methodsfor using a classical computer to generate a sequence of operations thatcan be used to operate a quantum computer.

(B) Description of Related Art

Henceforth, we will allude to certain references by codes. Here is alist of codes and the references they will stand for.

-   -   Ref.Bac is D. Bacon, “How a Clebsch-Gordan transform helps to        solve the Heisenberg hidden subgroup problem”,        arXiv:quant-ph/0612107    -   Ref.BCH1 is D. Bacon, I. Chuang, and A. Harrow, “Efficient        Quantum Circuits for Schur and Clebsch-Gordan Transforms”,        arXiv:quant-ph/0407082    -   Ref.BCH2 is D. Bacon, I. Chuang, and A. Harrow, “The Quantum        Schur Transform: I. Efficient Qudit Circuits”,        arXiv:quant-ph/0601001    -   Ref.CSD is R. R. Tucci, “A Rudimentary Quantum Compiler (2cnd        Ed.)”, arXiv quant-ph/9902062    -   Ref.Cvi is Predrag Cvitanovic, Group theory: birdtracks, Lie's,        and exceptional groups (Princeton University Press, 2008) Also        available on the web.    -   Ref.Dev is S. Devitt, Kae Nemoto, and W. Munro, “Quantum error        correction for beginners”, arXiv:0905.2794    -   Ref.Har is A. Harrow, “Applications of coherent classical        communication and the Schur transform to quantum information        theory”, arXiv:quant-ph/0512255    -   Ref.Jak is P. Jakubczyk, Y. Kravets, D. Jakubczyk, “An        alternative approach to the construction of Schur-Weyl        transform”, arXiv:1409.6130    -   Ref.Kep is S. Keppeler, M. Sjodahl, “Hermitian Young Operators”,        arXiv:1307.6147    -   Ref.Ma is Zhong-Qi Ma, Group Theory for Physicists (World        Scientific, 2007)    -   Ref.NaMi is P. Nataf, F. Mila, “Exact Diagonalization of        Heisenberg SU(N) models”, arXiv:1408.5341 [quant-ph]    -   Ref.Paulinesia is R. R. Tucci, “QC Paulinesia”,        arXiv:quant-ph/0407215    -   Ref.RevKit is M. Soeken, S. Frehse, R. Wille, and R. Drechsler,        “RevKit: A Toolkit for Reversible Circuit Design”,        Multiple-Valued Logic and Soft Computing 18, no. 1 (012): 55-65.    -   Ref.Schen is Irene Verona Schensted, A course on the application        of group theory to quantum mechanics (Peaks Island, Me.: Neo        Press, 1976)    -   Ref.Tuc-HYP is R. R. Tucci, “Quantum Circuit For Unitary Matrix        that Simultaneously Diagonalizes Set of Hermitian Young        Projection Operators”. Unpublished. Copy included as an appendix        to Ref.Tuc-HYP-PAT.    -   Ref.Tuc-HYP-PAT is R. R. Tucci, “Method For Projecting Out        Irreducible Representations from a Quantum State Of Multiple        Qubits”, patent application Ser. No. 14/499,539    -   Ref.TucOpAv is R. R. Tucci, “Method for Evaluating Quantum        Operator Averages”, U.S. Pat. No. 8,612,499    -   Ref.Tuc-PRISM is R. R. Tucci, “How to Calculate and Compile the        Irreps Generator for n Qu(d)its”. Unpublished. Copy included as        an appendix to this patent application.

This invention deals with quantum computing. A quantum computer is anarray of quantum bits (qubits) together with some hardware formanipulating those qubits. Quantum computers with several hundred qubitshave not been built yet. However, once they are built, it is expectedthat they will perform certain calculations much faster that classicalcomputers. A quantum computer follows a sequence of elementaryoperations. The operations are elementary in the sense that they act ononly a few qubits (usually 1, 2 or 3) at a time. Henceforth, we willsometimes refer to sequences as products and to operations as operators,matrices, instructions, steps or gates. Furthermore, we will abbreviatethe phrase “sequence of elementary operations” by “SEO”. SEOs forquantum computers are often represented by quantum circuits. In thequantum computing literature, the term “quantum algorithm” usually meansa SEO for quantum computers for performing a desired calculation. Somequantum algorithms have become standard, such as those due toDeutsch-Jozsa, Shor and Grover. One can find on the Internet manyexcellent expositions on quantum computing.

By compiling a unitary matrix, we mean expressing it as a SEO (a.k.a, aquantum circuit).

This invention gives an application of Group Theory (or, morespecifically, Group Representation Theory) to quantum computing.

Group Representation Theory is of great importance in Physics. For anon-rigorous, addressed mainly to physicists, introduction to GroupRepresentation Theory, see, for example, the textbooks Ref.Schen,Ref.Ma, Ref.Cvi. Out of those textbooks, Ref.Cvi by Cvitanovic is uniquein that it is the only one of the three that uses diagrammatic notationthat Cvitanovic calls “birdtracks”. Cvitanovic was not the first to usebirdtracks, but he greatly advanced and popularized their use in GroupTheory. Birdtracks are closely related to another diagrammatic notation,the quantum circuits used in quantum computing. This invention tries tomerge these 2 types of diagrammatic notations.

A very important tool in Group Representation Theory is that of Youngdiagrams and their associated Young Projectors. Young diagrams have along, illustrious history. They are discussed at length in the textbooksRef.Schen, Ref.Ma, Ref.Cvi.

Young projectors are not generally Hermitian, but it is possible toconstruct Hermitian operators that project out the same subspaces as theusual Young projectors. We shall call such operators HYPs (HermitianYoung Projectors). Keppeler and Sjodahl were the first to give, inRef.Kep, a method for constructing HYPs, together with a proof of thecorrectness of their method. In this invention, the Keppeler, Sjodahlmethod for constructing HYPs will be used frequently.

In this patent, the letters n and d will be reserved to denote thefollowing. If d=2, 3, 4, . . . , then qu(d)its are the fundamentalparticles or quarks of U(d) (the set of d×d unitary matrices). Qu(d)itshave d orthonormal states labeled |j> with j∈{0, 1, 2, . . . , d−1}. Aqu(d)it lives in the complex span of these d states. The d in U(d) isoften called the number of colors. Qu(d)its with d=2 are called qubits.Each horizontal line or wire in a quantum circuit carries one qu(d)it.The letter n is often used to denote the number of horizontal lines inthe quantum circuit. In this patent, the same number n will also bereferred to as the number of particles or qu(d)its.

Since all the HYPs for a fixed (n, d) are mutually orthogonal, they willalso commute with each other. By definition, the HYPs are all Hermitian.It follows that one can find a transformation U that diagonalizes atonce all of the HYPs for fixed (n, d). We shall refer to this U as theIrreps Generator (or Irreps Gen for short) for n qu(d)its.

This patent builds on the results of Ref.Tuc-HYP-PAT, which is aprevious patent application also by Tucci. Ref.Tuc-HYP-PAT gives amethod of calculating and compiling an Irreps Gen for an arbitrarynumber n of particles but only for d=2 colors (qubits). Unfortunately,the method of Ref.Tuc-HYP-PAT does not generalize easily to arbitrary nand d (qu(d)its).

The preferred embodiment of this invention is a method for calculatingand compiling an Irreps Gen U for arbitrary (n, d). Thus, the method ofthis patent is more general than that of Ref.Tuc-HYP-PAT in the valuesof d to which it applies. Furthermore, the methods of both patents aresignificantly different, even for d=2.

Whereas in Ref.Tuc-HYP-PAT, we referred to U as the stabsXqu(d)its orquantum Young transform, in this patent we change its name to the IrrepsGen.

To compile U in the preferred embodiment of this invention, we use theCS-Decomposition of Linear Algebra. This decomposition was first usedfor quantum computing in Ref.CSD. See Ref.CSD for references discussingthe history and theory behind the CS Decomposition.

To compile U in the preferred embodiment of this invention, we firstexpress it as a product of unitary matrices Λ_(μ) ⁵⁵⁴ , eachcorresponding to a different irrep μ. Then we apply the CS-Decompositionto each Λ_(μ) ⁵⁵⁴ separately. Since the method of compilation given inthis paper respects the symmetry of the problem by applying theCS-Decomposition to each irrep μ separately rather than to the whole Uat once, we expect that it will lead to much shorter SEOs than if themethod did not respect the symmetry of the problem.

FIG. 7 below is dedicated to explaining some of the many applications ofthe Irreps Gen.

Bacon, Chuang and Harrow have defined, in Ref.BCH1, Ref.BCH2, Ref.Har,and Ref.Bac, two transforms that they call the Clebsch Gordan (CG) andSchur transforms. When n=2, the Schur and CG transforms are equal. Forarbitrary n, the Schur transform is a chain of CG transforms. InRef.Tuc-HYP, we explain several significant differences between ourIrreps Gen and their Schur transform.

In Ref.Jak, Jakubczyk et al give an “alternative approach” tocalculating the same transform as Bacon, Chuang and Harrow. They don'tmention the Keppeler, Sjodahl HYPs.

BRIEF SUMMARY OF THE INVENTION

The usual Young projectors are not generally Hermitian, but, inspired byconjectures of previous workers, Keppeler and Sjodahl showed in 2013 howto construct Hermitian operators that project out the same subspaces asthe usual Young projectors. We shall call such operators HYPs (HermitianYoung Projectors). All HYPs for a fixed number n of qu(d)its aremutually commuting and Hermitian so there exists a unitary matrix thatdiagonalizes all of them at once. We call such a unitary matrix anIrreps Generator or Irreps Gen for short. The invention presented hereis a method of constructing a quantum circuit for the Irreps Gen forarbitrary (n, d), where n is the number of particles or qu(d)its and dis the number of colors.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a block diagram of a classical computer feeding data to aquantum computer.

FIG. 2 shows some nonstandard notation used in this patent. For example,this figure shows how we use projectors to indicate the partitioning ofa matrix into 4 sub-blocks.

FIG. 3 shows how we define J-properties and H-properties, and how onecan express the Irreps Gen U as a sum of operators with each summandcorresponding to a different irrep μ.

FIG. 4 shows how to calculate a J_(μ), and a D_(μ) for each μ=1, 2, . .. , N_(ir).

FIG. 5 shows how to express the Irreps Gen U as a product ofμ-Householder Transformations At where Λ_(μ) ⁵⁵⁴ =1, 2, . . . , N_(ir).

FIG. 6 shows how to express the Irreps Gen U as a quantum circuit byapplying the CS-decomposition to each Λ_(μ) ⁵⁵⁴ obtained in FIG. 5.

FIG. 7 shows two applications of the Irreps Gen U, (a) to produce exoticCNOTs and (b) to express conveniently an arbitrary qu(d)it permutation.

DETAILED DESCRIPTION OF THE INVENTION

This section describes in detail a preferred embodiment of the inventionand other possible embodiments of the invention. For extra details aboutthis invention, see Ref.Tuc-HYP, Ref.Tuc-HYP-PAT, Ref.Tuc-PRISM andreferences therein.

(A) Preferred Embodiment

FIG. 1 is a block diagram of a classical computer feeding data to aquantum computer. Box 100 represents a classical computer. The IrrepsGen SEO instruction set is generated inside Box 100. Box 100 comprisessub-boxes 101, 102, 103. Box 101 represents input devices, such as amouse or a keyboard. Box 102 comprises the CPU, internal and externalmemory units. Box 102 does calculations and stores information. Box 103represents output devices, such as a printer or a display screen. Box105 represents a quantum computer, comprising an array of quantum bitsand some hardware for manipulating the state of those bits.

Next consider FIG. 2.

For integers a, b such that a≦b, define Z_(a,b) by 201.

Let N_(st)=d^(n) be the number of states for n qu(d)its. We will oftenrefer to n as the number of particles or qu(d)its and to d as the numberof colors.

As indicated by 202, the HYPs H_(μ) are labeled by the elements ofstabs(n, d). Here stabs(n, d) is the set of all standard Young Tableauwith n boxes and ≦d rows. stabs(n, d) is also the set of n particleirreps μ of U(d). See Ref.Tuc-HYP for more information about the setstabs(n, d). In this patent, as indicated by 203, we will often identifythe sets stabs(n, d) and Z_(1,N) _(ir) , where N_(ir) equals the numberof irreps (the cardinality of stabs(n, d)).

Let |e_(k)> for k=1, 2, . . . , N_(st) denote an orthonormalN_(st)-dimensional basis. Let A be a subset of Z_(1,N) _(st) and A^(c)the complement of A in Z_(1,N) _(st) . Then J_(A) and its complementJ_(A) ^(c) are defined by 204 and 205, respectively.

We will write M_(J) _(A) _(,J) _(B) for the projection 206 of a generalN_(st)×N_(st) matrix M.

If J=J_(A) for some A⊂Z_(1,N) _(st) and M is an N_(st)×N_(st) matrix, wedefine by 207 the “J-block inverse of M”, assuming it exists. Here Ω⁻¹denotes the usual inverse of a square matrix Ω. Clearly, the J-blockinverse of M inverts only the sub-block of M given by its J rows andcolumns.

If J=J_(A) for some A⊂Z_(1,N) _(st) and M is an N_(st)×N_(st) matrix, wewill often symbolically partition M into the 4 quadrants correspondingto the J and J^(c) rows and columns of M. We will do so by writing 208.

Note that in 208, [[·]] does not stand for the conventional matrixsymbol since all the entries inside the [[·]] are conventionalN_(st)×N_(st) matrices themselves. As indicated by 208, the [[·]] symbolis just a convenient way of organizing 4 matrices of the same size andindicating that all 4 should be summed together. For the remainder ofthis patent, we will be lazy and use [·] to indicate both [[·]] and theconventional [·]. Which one we are referring to should be clear to thereader from context.

Next consider FIG. 3.

Consider a set {J_(μ): μ=1, 2, . . . N_(ir)} of matrices (or associatedlinear operators), all square and of the same size. We will say thatsuch a set satisfies the J-properties if for all μ and ν, 301 issatisfied.

Consider a set {H_(μ): μ=1, 2, . . . N_(ir)} of matrices (or associatedlinear operators), all square and of the same size. We will say thatsuch a set satisfies the H-properties if for all μ and ν, 302 issatisfied.

We will say that an operator A lives in irrep μ (with respect to a setof J_(μ), operators that satisfies the J-properties) if 303 issatisfied. We will consider a set of operators D_(μ) that satisfies 304for all μ.

Suppose U is a unitary matrix that can be expressed as 305, where theH_(μ) satisfy the H-properties and D_(μ) lives in μ for all μ. Then wewill refer to U as the Irreps Gen (Irreducible representationsGenerator). In this patent, we will use the letter U exclusively forthat transform, except when we use it for U(d), the group of d×d unitarymatrices. Note that there is a different U for each (n, d) pair.

Next consider FIG. 4.

Ref.Kep gives us a method for calculating a set of HYPs H_(μ) thatsatisfies the H-properties. FIG. 4 gives an algorithm such that, given aset of H_(μ) that satisfies the H-properties, we can do all of thefollowing: compute a set of J_(μ) that satisfies the J-properties, and aset of D_(μ), such that for all μ, D_(μ) lives in μ, andU=Σ_(μ)H_(μ)D_(μ) is unitary.

Next we will describe very briefly the algorithm of FIG. 4. For a moredetailed description of the algorithm, see Ref.Tuc-PRISM.

Let X_(μ) be defined by 401, where L2M( ) is a function that translatesLists to Matrices by taking its list argument and returning the diagonalmatrix with that list argument as diagonal. Find as many x_(k) ^(μ) aspossible such that 402 and 403 are satisfied for all μ, ν. It'sconvenient to define x_(k) by 404, where μ(k) is the single μ for whichx_(k) ^(μ)≠0. Using the information collected so far for the J_(μ), finda full set of J_(μ) while being mindful of satisfying the J-propertiesand the constraint that tr(J_(μ))=tr(H_(μ)) for all μ.

At this point, we should have H_(μ) and J_(μ) for all μ. Using thatinformation, one can find the D_(μ) for each μ via the equations 405 and406, where U_(μ) is unitary and X_(μ) is defined by 401.

Next consider FIG. 5.

In FIG. 5, we define a generalization of the conventional HouseholderTrans-formations (HTs) of Linear Algebra. We will call suchgeneralizations μ-HTs. A conventional HT, when applied to a unitarymatrix U, diagonalizes a single column of U. By “diagonalizes” a columnof U, we mean that it makes all entries of the column equal to zeroexcept for one of them. A μ-HT, on the other hand, diagonalizes multiplecolumns of U at once; more precisely, it diagonalizes all columns of Ucorresponding to the irrep μ.

Performing Algorithm 2 labeled 502 generates a sequence of μ-HTs Λ_(μ)such that 501 is satisfied. (We will also refer to Λ_(μ) ⁵⁵⁴ as a μ-HT).Since the Λ_(μ) are unitary, 501 can be easily inverted to express theIrreps Gen U as a product of Λ_(μ) ⁵⁵⁴ (times a trivial phase factor).Note how equation 503 expresses each Λ_(μ) ⁵⁵⁴ as a matrix partitionedinto four quadrants given by the J_(μ) and J_(μ) ^(c) rows and columns.We refer the reader to Ref.Tuc-PRISM for more details about thealgorithm of FIG. 5.

Next consider FIG. 6.

In the previous figure, we showed how the Irreps Gen U can be expressedas a product of μ-HTs. In this figure, we show how to compile each ofthese μ-HT factors using the well-known (See Ref.CSD) CS-Decompositionof Linear Algebra.

The CS decomposition is performed on a unitary matrix which has beenpartitioned into four quadrants. At the same time, the Λ_(μ) ⁵⁵⁴generated by Algorithm 2 of FIG. 5 are unitary matrices which are in theform 503, so they are a priori partitioned into four quadrants given bythe J_(μ) and J_(μ) ^(c) rows and columns. Thus, doing aCS-decomposition of a Λ_(μ) ⁵⁵⁴ with the predefined partitioning is verynatural and it yields 601. In that figure, the diagonal blocks of

_(μ) and

_(μ) are unitary. Furthermore, all quadrants of

_(μ) are diagonal and satisfy other constraints that are described infull generality in Ref.Tuc-PRISM. See that reference for more details.

Here is an example. If tr(J_(μ))=2 and tr(J_(μ) ^(c))=3, then

_(μ),

_(μ) and

_(μ) can be found of the form given by Box 602. If the 2 and 3 for thetrace values are swapped, then we get 603 instead of 602. Note that 604applies to both cases 602 and 603.

For each μ, after doing a CS-Decomposition of Λ_(μ) ⁵⁵⁴ , one can usethe techniques described in Ref.CSD or similar ones to decompose theresulting matrices

_(μ),

_(μ),

_(μ) into a SEO (sequence of elementary transformations).

Next consider FIG. 7.

FIG. 7 describes some possible applications of the invention. Thisfigure is almost the same as FIG. 6 of Ref.Tuc-HYP-PAT, but it has beenadapted so as to apply to arbitrary d instead of to only d=2.

One important application of the Irreps Gen is that it allows us tocreate a new kind of CNOT. The usual CNOT is 701 where τ is the targetqubit, κ is the control qubit, σ_(x) is the X Pauli matrix, b∈{0,1} andP_(b)=|b><b|. By virtue of UJ_(μ)U^(†)=H_(μ), one can construct theexotic singly controlled qu(d)it rotation given by 702, where τ is thetarget qu(d)it, and κ^(n) are n control qu(d)its. Also, Γ(τ) is anelement of U(d) acting on τ and H_(μ)(κ^(n)) with μ∈stabs(n, d) is a HYPacting on κ^(n). The U(κ^(n)) is the Irreps Gen acting on κ^(n).

One can also generalize a multiply controlled CNOT given by 703, where τis the target qubit, κ_(l) is a control qubit which is different fordifferent l, and b_(l)∈{0, 1}, to an exotic multiply controlled qu(d)itrotation given by 704, where τ is the target qu(d)it and κ_(l) ^(n) ^(l)are n_(l) control qu(d_(l))its which are different particles fordifferent l. Also, Γ(τ) is an element of U(d) acting on τ and H_(μ) _(l)(κ_(l) ^(n) ^(l) ) with μ_(l)∈stabs(n_(l), d_(l)) is a HYP acting onκ_(l) ^(n) ^(l) . The U(κ_(l) ^(n) ^(l) ) are the Irreps Gen acting onκ_(l) ^(n) ^(l) .

HYPs also lead to a new, fairly simple matrix representation ofarbitrary qu(d)it permutations. Next, we will say something about thisnew representation. See Ref.Tuc-HYP for more details for the case d=2.This new representation was inspired by Ref.NaMi written by Natal andMila. They too are concerned with finding a convenient matrixrepresentation for arbitrary qu(d)it permutations. Whereas we use HYPs,they use instead something invented by Young called “orthogonal units”.

Suppose μ∈stabs(n, d). Let |μ,α> where α=1, 2, . . . , dim(μ) be a setof orthonormal eigenvectors with eigenvalue 1 of H_(μ). Then the statesin the set B defined by 705 are orthonormal. Furthermore, if B′ is givenby 706, then the complex span of B equals the complex span of B′. Call Bthe HYP basis. In the HYP basis, H_(μ) equals 707.

Let 708 denote a qu(d)it-swap of adjacent horizontal lines or qu(d)its jand j+1 in a qu(d)it circuit with horizontal lines labeled from top tobottom 0, 1, 2, . . . , n−1. Using the fact that Σ_(μ)H_(μ)=1, we get709. By the Wigner-Eckart theorem, we expect 710, where f_(μ, ν) ^(j) isindependent of α, and it vanishes if μ

ν (i.e., if μ and ν are not equivalent irreps).

For all μ,ν∈stabs(n, d), define H_(μ,ν) by 711, where θ(μ˜ν) equals 1 ifμ and ν are equivalent irreps, and it equals 0 otherwise. If we defineν_(n) by 712, then Ref.Tuc-HYP shows that any permutation of n qu(d)itsis an element of ν_(n).

(B) Other Embodiments

The preferred embodiment of this invention is a method for calculatingand compiling the Irreps Gen U for arbitrary (n, d).

It's important to note that most of the algorithms of this patent do notassume that the set of H_(μ) is necessarily a set of HYPs; they onlyassume that it satisfies the H-properties. Thus, the methods of thispaper can also be applied to other sets of H_(μ) operators, as long asthey satisfy the H-properties.

A standard definition in the field of quantum computation is that aqu(d)it is a quantum state that belongs to a d dimensional vector spaceand a qubit is a qu(d)it with d=2. In quantum error correction (seeRef.Dev for an introduction), one distinguishes between 2 types ofqu(d)its, physical and logical. A logical qu(d)it is mapped into anumber of physical qu(d)its. It goes without saying that the qu(d)its inthe quantum circuit of Irreps Gen proposed in this invention can alwaysbe interpreted as logical qu(d)its, and additional gates can be added tothat quantum circuit with the purpose of performing error correction.

By compiling U we mean expressing it as a SEO (sequence of elementaryoperations) on qubits, not qu(d)its, even in the case when the HYPs arefor projecting out irreps of n qu(d)its with d≠2 colors. When d≠2, wewill say that the HYP qu(d)its are meta-logical, whereas the qubits intowhich we compile them are logical. If quantum error correction is added,the logical qubits might themselves be encoded into physical qubits, orelse the meta-logical qu(d)its might be encoded directly into physicalqubits.

For convenience, the quantum circuit for the Irreps Gen proposed in thisinvention may include gates that act on more than 3 qubits at a time.Such “fat” gates might be judged by some not to be elementary gates asdefined earlier in this patent. However, such fat gates should beallowed inside the SEO's covered by this invention for cases in whichthey are trivially expandable (TE) fat gates. By TE fat gates we mean,fat gates for which there are well known, expanding methods forreplacing them by a sequence of gates that are strictly elementary, inthe sense that they act on just one or two qubits at a time.Multi-controlled rotations and multiplexors are examples of TE fatgates. In fact, see the Java classes MultiCRotExpander andMultiplexorExpander and related classes available at www.ar-tiste.comand as a computer code appendix to several Tucci patents, for exampleRef.TucOpAv. These classes automate such expanding methods formulti-controlled rotations and multiplexors.

A promising avenue for future research is to modify our exact method forcomputing U and seek a bounded error approximation for U and for itscompilation, some sort of expansion in powers of 1/n useful when n>>1.That such an approximation is likely to exist can be inferred from thefact that for d=2 and large n, what we are dealing with is a system ofn>>1 spin ½ quantum states, a system which should behave approximatelyas if it had classical angular momentum J=n/2.

So far, we have described some exemplary preferred embodiments of thisinvention. Those skilled in the art will be able to come up with manymodifications to the given embodiments without departing from thepresent invention. Thus, the inventor wishes that the scope of thisinvention be determined by the appended claims and their legalequivalents, rather than by the given embodiments.

I claim:
 1. A method of operating a classical computer to calculate atotal SEO, with the purpose of using said total SEO to operate a quantumcomputer, and to induce said quantum computer to undergo a unitarytransformation U that acts on n qu(d)its, wherein U simultaneouslydiagonalizes a set of operators H_(μ), that satisfies the H-properties,wherein said U can be expressed in the form${U = {\sum\limits_{\mu}\; {H_{\mu}D_{\mu}}}},$ wherein each operatorD_(μ) lives in μ with respect to a set of J_(μ) operators that satisfiesthe J-properties, wherein said total SEO comprises a product of unitaryoperators Λ_(μ) ⁵⁵⁴ , wherein each Λ_(μ) is designed to diagonalize forthe matrix to which it is applied, all rows or columns corresponding toindex μ.
 2. The method of claim 1, wherein said set of H_(μ) is a set ofHYPs (Hermitian Young Projectors) and the index μ ranges over stabs(n,d) or a set with the same cardinality, wherein stabs(n, d) is the set ofall Young Tableau with n boxes and ≦d rows, wherein H_(μ) projects outthe n particle irreducible representation μ of U (d).
 3. The method ofclaim 1, whereinUJ_(μ)U^(†)−H_(μ) for all μ.
 4. The method of claim 1, wherein saidtotal SEO induces said quantum computer to undergo an exotic controlledqu(d)it rotation R equal toΓ(τ)^(H) ^(μ) ^((κ) ^(n) ⁾, wherein τ is the target qu(d)it and κ^(n)are n control qu(d)its, wherein Γ(τ) is an element of U(d) acting on τ,wherein said unitary transformation U and its Hermitian conjugate U^(†)are sub-operations of said R, and said sub-operations act on the κ^(n).5. The method of claim 1, wherein for each μ, Λ_(μ) ⁵⁵⁴ is decomposedinto 3 factors as prescribed by the CS-Decomposition.
 6. A device thatcalculates a total SEO, with the purpose of using said total SEO tooperate a quantum computer, and to induce said quantum computer toundergo a unitary transformation U that acts on n qu(d)its, wherein Usimultaneously diagonalizes a set of operators H_(μ) that satisfies theH-properties, wherein said U can be expressed in the form${U = {\sum\limits_{\mu}\; {H_{\mu}D_{\mu}}}},$ wherein each operatorD_(μ) lives in μ with respect to a set of J_(μ) operators that satisfiesthe J-properties, wherein said total SEO comprises a product of unitaryoperators Λ_(μ) ⁵⁵⁴ , wherein each Λ_(μ) is designed to diagonalize forthe matrix to which it is applied, all rows or columns corresponding toindex μ.
 7. The device of claim 6, wherein said set of H_(μ) is a set ofHYPs (Hermitian Young Projectors) and the index μ ranges over stabs(n,d) or a set with the same cardinality, wherein stabs(n, d) is the set ofall Young Tableau with n boxes and ≦d rows, wherein H_(μ) projects outthe n particle irreducible representation μ of U(d).
 8. The device ofclaim 6, wherein said total SEO induces said quantum computer to undergoan exotic controlled qu(d)it rotation R equal toΓ(τ)^(H) ^(μ) ^((κ) ^(n) ⁾, wherein τ is the target qu(d)it and κ^(n)are n control qu(d)its, wherein Γ(τ) is an element of U(d) acting on τ,wherein said unitary transformation U and its Hermitian conjugate U^(†)are sub-operations of said R, and said sub-operations act on the κ^(n).9. The device of claim 6, wherein for each μ, Λ_(μ) ⁵⁵⁴ is decomposedinto 3 factors as prescribed by the CS-Decomposition.
 10. A method ofoperating a classical computer to calculate a total SEO, with thepurpose of using said total SEO to operate a quantum computer, and toinduce said quantum computer to undergo a transformation${A\; {\Gamma (\tau)}{\prod\limits_{ = 1}^{L}\; {\left\{ {J_{\mu_{}}\left( \kappa_{}^{n_{}} \right)} \right\} A^{\dagger}}}},{wherein}$${A = {\prod\limits_{ = 1}^{L}\; \left\{ {U_{}\left( \kappa_{}^{n_{}} \right)} \right\}}},$wherein τ is the target qu(d)it and κ_(l) ^(n) ^(l) are n_(l) controlqu(d_(l)) its which are different particles for different l, whereinΓ(τ) is an element of U(d) acting on τ, wherein J_(μ) _(l) (κ_(l) ^(n)^(l) ) for all μ_(l) with fixed l is a set of operators satisfying theJ-properties and acting on κ_(l) ^(n) ^(l) , wherein U_(μ) _(l) (κ_(l)^(n) ^(l) ) is a unitary operator acting on κ_(l) ^(n) ^(l) andsatisfyingU _(μ)(α)J _(μ)(α)U _(μ)(α)^(†) =H _(μ)(α) with μ=μ_(l) and α=κ_(l) ^(n)^(l) , wherein H_(μ) _(l) (κ_(l) ^(n) ^(l) ) for all μ_(l) with fixed lis a set of operators satisfying the H-properties, wherein said U_(μ)_(l) (κ_(l) ^(n) ^(l) ) comprises a product of unitary operators Λ_(μ)_(l) ⁵⁵⁴ for all μ_(l) at fixed l, wherein each Λ_(μ) _(l) is designedto diagonalize the J_(μ) _(l) rows or columns of some matrix.
 11. Themethod of claim 10, wherein said set of H_(μ) _(l) (κ_(l) ^(n) ^(l) )for all μ_(l) with fixed l is a set of HYPs (Hermitian Young Projectors)and the index μ_(l) ranges over stabs(n_(l),d_(l)) or a set with thesame cardinality, wherein stabs(n_(l),d_(l)) is the set of all YoungTableau with n_(l) boxes and ≦d_(l) rows, wherein H_(μ) _(l) (κ_(l) ^(n)^(l) ) projects out the n_(l) particle irreducible representation μ_(l)of U(d_(l)).
 12. A device that calculates a total SEO, with the purposeof using said total SEO to operate a quantum computer, and to inducesaid quantum computer to undergo a transformation${A\; {\Gamma (\tau)}{\prod\limits_{ = 1}^{L}\; {\left\{ {J_{\mu_{}}\left( \kappa_{}^{n_{}} \right)} \right\} A^{\dagger}}}},{wherein}$${A = {\prod\limits_{ = 1}^{L}\; \left\{ {U_{}\left( \kappa_{}^{n_{}} \right)} \right\}}},$wherein τ is the target qu(d)it and κ_(l) ^(n) ^(l) are n_(l) controlqu(d_(l)) its which are different particles for different l, whereinΓ(τ) is an element of U(d) acting on τ, wherein J_(μ) _(l) (κ_(l) ^(n)^(l) ) for all μ_(l) with fixed l is a set of operators satisfying theJ-properties and acting on κ_(l) ^(n) ^(l) , wherein U_(μ) _(l) (κ_(l)^(n) ^(l) ) is a unitary operator acting on κ_(l) ^(n) ^(l) andsatisfyingU _(μ)(α)J _(μ)(α)U _(μ)(α)^(†) =H _(μ)(α) with μ=μ_(l) and α=κ_(l) ^(n)^(l) , wherein H_(μ) _(l) (κ_(l) ^(n) ^(l) ) for all p with fixed £ is aset of operators satisfying the H-properties, wherein said U_(μ) _(l)(κ_(l) ^(n) ^(l) ) comprises a product of unitary operators Λ_(μ) ⁵⁵⁴for all μ_(l) at fixed l, wherein each Λ_(μ) _(l) , is designed todiagonalize the J_(μ) _(l) rows or columns of some matrix.
 13. Thedevice of claim 12, wherein said set of H_(μ) _(l) (κ_(l) ^(n) ^(l) )for all μ_(l) with fixed l is a set of HYPs (Hermitian Young Projectors)and the index μ_(l) ranges over stabs(n_(l),d_(l)) or a set with thesame cardinality, wherein stabs(n_(l),d_(l)) is the set of all YoungTableau with n_(l) boxes and ≦d_(l) rows, wherein H_(μ) _(l) (κ_(l) ^(n)^(l) ) projects out the n_(l) particle irreducible representation μ_(l)of U(d_(l)).